Soc Choice Welfare (2006) 27: 545–570 DOI 10.1007/s00355-006-0126-y

O R I G I NA L PA P E R

Carmelo Rodríguez-Álvarez

Candidate stability and voting correspondences

Received: 7 February 2003 / Accepted: 7 September 2005 / Published online: 9 June 2006 © Springer-Verlag 2006

Abstract We extend the analysis of Dutta et al. (in Econometrica, 69:1013–1038, 2001) on strategic candidacy to multivalued environments. For each agenda and each profile of voters’ preferences over running candidates, a voting correspondence selects a set of running candidates. A voting correspondence is candidate stable if no candidate ever has an incentive to withdraw her candidacy when all other potential candidates run for office. In the multivalued framework, candidates’ incentives to withdraw depend on candidates’ preferences over sets. If candidates cannot vote and they compare sets of candidates according to their expected utility This paper is a revised version of the second chapter of my Ph.D. Dissertation submitted to the Universitat Autònoma de Barcelona. I am indebted to my supervisor Salvador Barberà for his advice, encouragement, and patience. I thank the hospitality of the Wallis Institute of Political Economy at the University of Rochester, where the revision of this paper was conducted. I am grateful to two anonymous referees and the Associate Editor, John Weymark, for their exhaustive and insightful comments. I also thank Dolors Berga, Carmen Beviá, Walter Bossert, Jernej ˇ c, Bhaskar Dutta, Matt Jackson, Jordi Massó, Diego Moreno, David Pérez-Castrillo, and Copiˇ Yves Sprumont for helpful conversations and suggestions. Financial support through Research Grant 1998FI00022 from Comissionat per Universitats i Recerca, Generalitat de Catalunya, Research Project PB98-870 from the Ministerio de Ciencia y Tecnología, Fundación Barrié de la Maza, and Consejería de Innovación, Ciencia y Empresa, Junta de Andalucía is gratefully acknowledged. The usual disclaimer applies. C. Rodríguez-Álvarez Departamento de Economía, Universidad Carlos III de Madrid, 28903 Getafe (Madrid), Spain C. Rodríguez-Álvarez (B) Departamento de Teoría e Historia Económica, Universidad de Málaga. Plaza El Ejido s/n Ap. Of. Suc. 4, 29071 Málaga, Spain E-mail: [email protected]

546

C. Rodríguez-Álvarez

conditional on some prior probability assessment, then a voting correspondence satisfies candidate stability and unanimity if and only if it is dictatorial. If the probability assessments are restricted to be uniform, candidates’ preferences over sets are consistent with leximin preferences, or candidates can vote, then possibility results are obtained. 1 Introduction The decision of a candidate to run for office or not may be of capital importance for the result of an election. Of course, candidates must run to win the election. But, in fact, the presence (or the absence) of a candidate can affect the final result of the election, even if that candidate has no chance of winning. In a recent paper, Dutta et al. (2001) (henceforth DJL) initiate the study of candidates’ incentives to manipulate the result of an election by withdrawing.1 DJL analyze deterministic voting procedures that select a winning candidate for each set of running candidates and each profile of voters’ preferences over running candidates. They introduce a stability condition called candidate stability. A voting procedure is candidate stable if no candidate has an incentive to withdraw when all other potential candidates run for office. Candidate stability incorporates the idea that the set of running candidates is the result of a Nash equilibrium. Assuming that candidates cannot vote, DJL show that in their single-valued framework, only dictatorships satisfy candidate stability and unanimity. If candidates can vote, then candidate stable and unanimous voting procedures do exist. However, every candidate stable voting procedure fails to satisfy minimal regularity conditions. In the present work, we generalize DJL’s analysis to multivalued voting rules. Therefore, we model elections as voting correspondences. For each set of running candidates and each profile of voters’ preferences over running candidates, a voting correspondence selects a set of running candidates. A voting correspondence can be interpreted as a first screening device that narrows the social agenda to a smaller set of candidates. Candidates know this first selection but they are not aware of the final resolution of the social choice. Therefore, candidates consider the selected set as the result of the election. When the result of the social choice is multivalued, the study of the candidates’ strategic concerns becomes problematic since preferences over candidates do not contain enough information to compare sets of candidates. We explore two alternative approaches. First, we assume that candidates are equipped with preferences over sets of candidates. These preferences over sets are supposed to be consistent with some initial preferences over candidates.2 We analyze candidates’ incentives when candidates’ preferences over sets are consistent with expected utility maximization. We also study domains of preferences over sets that are consistent with extreme attitudes toward risk, such as leximin preferences. Alternatively, we propose two stability conditions that incorporate the idea that candidates cannot 1 Other important articles analyze candidates’ incentives for specific voting rules. For instance, Osborne and Slivinski (1996) and Besley and Coate (1997) concentrate on large elections with plurality rule and ideological positions of the candidates. On the other hand, Dutta et al. (2002) examine strategic candidacy in the context of voting by successive elimination. 2 See Barberà et al. (2004) for a recent survey on the topic of extending preferences over objects to preferences over sets of objects.

Candidate stability and voting correspondences

547

benefit from withdrawing. A first condition, no-harm, requires that if a candidate is elected when all potential candidates run, then she remains elected when some candidate withdraws. A second condition, insignificance, says that the withdrawal of a candidate who is not elected when all the candidates run has no effect on the election. Indeed, we see that, when candidates cannot vote and under different domains of preferences, candidate stability implies no harm and insignificance. We show that if candidates cannot vote and they compare sets according to their expected utility conditional on some prior probability assessment, then only dictatorial voting correspondences are candidate stable and unanimous. If the prior assessments are restricted to be uniform, then voting correspondences that select the best candidates for two fixed agents also satisfy both conditions. If candidates’ preferences are consistent with the leximin preferences, then more positive results are obtained. However, the power of decision remains concentrated in the hands of an arbitrary group of voters. Similar results apply if we investigate the implications of no harm and insignificance when candidates can vote. Before proceeding with the formal analysis, we review the most related literature. Eraslan and McLennan (2004) independently analyze the implications of candidate stability for multi-valued voting procedures when voters can express weak preferences over candidates. These authors do not explicitly model candidates’ incentives to withdraw, since candidates are not equipped with preferences over sets of candidates. Instead, they propose a strong candidate stability condition for multi-valued environments that is stronger than no harm and insignificance. In fact, their strong candidate stability is equivalent to candidate stability when candidates cannot vote and their preferences over sets are consistent with expected utility maximization conditional on an unrestricted prior probability assessment. Their main result is that only serially dictatorial rules satisfy strong candidate stability and unanimity. Also related is Ehlers and Weymark (2003). They provide a direct proof of DJL’s main result. In addition, they show that their proof can also be applied to multivalued environments to prove Eraslan and Mclennan’s (2004) result. A companion paper, Rodríguez-Álvarez (2006), considers the analysis of candidates’ incentives in a probabilistic framework. Elections are modeled as probabilistic voting procedures, that is, voting rules that for each agenda and each profile of preferences select a lottery on the set of candidates. It is shown that, when candidates cannot vote, only probabilistic versions of dictatorial voting procedures satisfy unanimity and never provide unilateral incentives for the candidates to withdraw given any set of potential candidates. Finally, we mention Berga et al. (2004) who study the problem of a society choosing from a set of potential new members. These authors focus on the existence of stable voting rules in the sense that founding members should not have incentives to leave the society if they do not like the resulting new society after the entry of new members. They show that a rule is stable and strategy-proof only if new members are admitted with the unanimous consent of the founding members.3 In contrast with candidate stability, their stability condition is not strategic. When considering whether or not to exit, a member does not take into account the effect of her withdrawal on the social outcome. Instead, she may consider withdrawing 3 A rule is strategy-proof if no founding member (voter) has an incentive to misrepresent her preferences.

548

C. Rodríguez-Álvarez

because she may prefer staying on her own rather than belonging to the resulting new society. The remainder of the paper is structured as follows. In Sect. 2, we introduce our notation and definitions. In Sect. 3, we present the implications of candidate stability when candidates cannot vote. In Sect. 4, we analyze the possible overlap between the sets of candidates and voters. In Sect. 5, we discuss further extensions and the relation between the notions of candidate stability and strategy-proofness. We gather all the proofs in the Appendices in Sect. 6. In Appendix 1, we include the proofs of the theorems. In Appendix 2, we prove the Lemmata that appear in Sect. 3.

2 Definitions 2.1 Voters, candidates, and preferences Let N be a finite society formed by a set of at least three candidates C , and a set of at least two voters V , N = C ∪ V , #C ≥ 3, and #V ≥ 2.4 An agenda A is a non-empty subset of C . Let 2C \{∅} denote the set of all agendas. Finally, let A denote the set of agendas that include all or all but one of the candidates, i.e. A ≡ {C } ∪ {C \{c} : c ∈ C } . A preference is a complete, antisymmetric, and transitive binary relation on C ∪ {∅}, where the empty-set refers to no-candidate being elected. Each i ∈ N is equipped with a preference Pi . We assume that for each a ∈ C , each i ∈ N , and each preference Pi , a Pi {∅}. Let P denote the set of all preferences. For each i ∈ N , P i ⊆ P denotes the domain of preferences over candidates admissible for i. For each i ∈ N , each A ∈ 2C \{∅}, and each Pi ∈ P i , top (A, Pi ) and bottom (A, Pi ) refer, respectively, to the first-ranked and the last-ranked candidates in A according to Pi . Preferences of voters who are not candidates are unrestricted, that is for each i ∈ N \ C , P i = P . However, we assume that each candidate considers herself as the best candidate. Hence, for each a ∈ C , P a = {Pa ∈ P | a = top(C , Pa )}. Let P r = ×i∈N P i . A preference profile P ∈ P r is a #N -tuple of preferences. For each A ∈ 2C \{∅} and each P ∈ P r , P | A denotes the restriction of P to the agenda A. Abusing notation, for each I ⊆ N , P I = ×i∈I P i , with typical element PI . We assume that candidates are equipped with preferences over sets of candidates. We analyze several domains of admissible preferences over sets, but we defer their presentation until the next section. A preference over sets of candidates  is a complete and reflexive binary relation on 2C \{∅}. Let D be the set of all preferences over sets of candidates. For each ∈ D,  refers to the strict component of , while ∼ refers to the indifference term that is defined from  in the standard way. We call a subset of preferences over sets of candidates DE ⊂ D a restricted domain of preferences over sets. We assume that candidates’ preferences over sets are obtained from their preferences over candidates. For each a ∈ C , each Pa ∈ P a , and each restricted domain DE , we denote by DaE (Pa ) ⊂ DE the set of 4

For each set S, #S stands for the cardinality of the set S.

Candidate stability and voting correspondences

549

Pa -consistent preferences over sets in the domain DE , that is DaE (Pa ) ≡ {∈ DE such that for each b, c ∈ C , {b}  {c} ⇔ b Pa c}.

Finally, DaE denotes the preferences over sets admissible for candidate a in the domain DE and is defined by DaE ≡ {∈ DE such that there is Pa ∈ P a with ∈ DaE (Pa )}.

2.2 Voting correspondences We are interested in voting rules that select a set of candidates for each agenda and each profile of voters’ preferences. A voting correspondence v is a mapping v : 2C \{∅} × P r → 2C \{∅} such that for each A ∈ 2C \{∅} and each P ∈ P r : (i) v (A, P) ⊆ A. (ii) For each P ∈ P r such that PV = PV , v(A, P) = v(A, P ).  (iii) For each P ∈ P r such that P | A = P | A , v (A, P) = v A, P . Part (i) refers to the fact that candidates cannot be elected if they are not at stake. Part (ii) implies that preferences of candidates who are not voters are not relevant for a voting correspondence. Part (iii) is referred to in the literature as independence of infeasible alternatives. It requires the choice set to be independent of preferences over candidates who do not run. Note that voting correspondences allow for multi-valued choices. Thus, voting correspondences are more general than a single-valued voting procedure as defined in DJL. Note also that, if the set of running candidates is fixed, then a voting correspondence becomes a social choice correspondence. 2.3 Candidate stability, no-harm, and insignificance We are interested in designing voting correspondences for which the agenda can be considered the result of candidates’ equilibrium decisions. Our main axiom introduces a minimal requirement of stability. Each candidate should prefer the outcome when all candidates run to the outcome that she would obtain if she withdrew. Of course, candidates’ incentives to withdraw depend on the particular domain of preferences over sets under consideration. Candidate stability in the domain DE (E -candidate stability). For each a ∈ C , each P ∈ P r , and each a ∈ DaE (Pa ), v (C , P) a v (C \{a}, P) . Next, we provide some conditions that incorporate the concept of candidate stability but that are independent of the specific domain of preferences. In the single-valued framework, by candidates’ self-preference, an elected candidate never has an incentive to withdraw. This is not the case in the multivalued framework. When a candidate is not the only one elected and her withdrawal results in the

550

C. Rodríguez-Álvarez

elimination of other candidates that she dislikes, but that were elected when all candidates run, she may prefer to withdraw rather than to run. Thus, we consider the following stability condition for elected candidates. If a candidate is selected when all the potential candidates run, then she remains elected when any other elected candidate withdraws. No-harm. For each a ∈ C and each P ∈ P r , a ∈ v(C , P) implies v(C , P) ⊆ v(C \{a}, P) ∪ {a}. We also introduce a stability condition that takes into account the possibility that a non-elected candidate withdraws. The withdrawal of a non-elected candidate never affects the outcome of the election. Insignificance. For each a ∈ C and each P ∈ P r , a ∈ / v(C , P) implies v(C , P) = v(C \{a}, P).5 2.4 Unanimity We assume that whenever all voters agree in who is the best running candidate, this candidate is elected. Note that if more than one running candidate is a voter, then such agreement cannot be attained. We employ a weak version of unanimity since candidate stability only applies to agendas in which no more than a single candidate withdraws. Unanimity. For each A ∈ A and each P ∈ P r , a = top (A, Pi ) for each i ∈ V implies v (A, P) = {a}. 3 Candidate stability when candidates cannot vote We start our analysis with the case in which voters and candidates’ sets do not overlap (C ∩ V = ∅). Although in many real life situations voters can run for office, this case may be a good approximation to settings in which the set of potential candidates is relatively small with respect to the set of voters. Hence, the capability of candidates to influence the result of the social choice as voters is almost negligible. In this section, we analyze the implications of candidate stability for different restricted domains of preferences over sets.

3.1 Candidate stability and conditional expected utility preferences We begin by introducing two domains of preferences over sets that were proposed in Barberà et al. (2001) for the study of strategy-proof social choice correspondences. Candidates are assumed to be equipped with von Neumann–Morgenstern 5 In the single-valued framework, when candidates cannot vote, DJL prove that candidate stability is equivalent to insignificance.

Candidate stability and voting correspondences

551

preferences over lotteries. If a set is selected, candidates attribute some probabilities of final choice to the elements of that set. Then, they compute the expected utility associated with that set. The proposed domains differ in the description of the way candidates assess the probabilities associated to sets. In the first domain, Conditional Expected Utility Consistent Preferences (or simply, BDS1), each candidate is assumed to have a single subjective probability distribution over C and to evaluate each subset according to its conditional expected utility. In the second domain, Conditional Expected Utility Consistent with Even Chance Probabilities Preferences (BDS2), the initial probability distribution is restricted to be uniform. Thus, every candidate associates each set with an even chance lottery over all its components. A utility function is a mapping u : C −→ R. Let i ∈ N and Pi ∈ P i . We say that u represents Pi if for each a, b ∈ C , a Pi b if and only if u (a)  > u (b). A probability assessment λi is a function λi : C −→ (0, 1) such that a∈C λi (a) = 1. Let a ∈ C and Pa ∈ P a . The preference over sets ∈ D is a Pa -consistent BDS1 preference – ∈ Da1 (Pa ) – if and only if there is a utility function u a representing Pa and a probability assessment λa such that for each X, Y ∈ 2C \{∅}: X Y ⇔

 x∈X

 λa (y) u a (y) λa (x) u a (x) ≥   . x ∈X λa (x ) λa (y ) y∈Y y ∈Y

Let a ∈ C and Pa ∈ P a . The preference over sets ∈ D is a Pa -consistent BDS2 preference – ∈ Da2 (Pa ) – if and only if there is a utility function u a representing Pa such that for each X, Y ∈ 2C \{∅}: X Y ⇔

1  1  u a (x) ≥ u a (y) . #X #Y x∈X

y∈Y

It is clear that BDS2 preferences are a special case of BDS1 preferences. For BDS2 preferences, prior probability assessments are restricted to be uniform. Thus, for each a ∈ C , Da2 ⊂ Da1 . Note that we rule out degenerate lotteries assigning null prior probability to some candidate as probability assessments. If we admitted degenerate lotteries as probability assessments, then we could not compute the conditional expected utility for some sets and we would obtain incomplete preferences over sets. Lemma 1 presents the main features of the preferences over sets that are admissible in both BDS domains. Lemma 1 Let a ∈ C and X, Y ∈ 2C \{∅}: (i) If a ∈ X \Y and X = (Y ∪ {a}), then for each ∈ Da1 , X  Y . (ii) If a ∈ X \Y and there is b ∈ Y \ X such that (X \{a}) = (Y \{b}), then for each ∈ Da2 , X  Y . (iii) If a ∈ / (X ∪ Y ) and X  = Y , then there exists ∈ Da2 such that X  Y . (iv) If a ∈ Y \ X , #Y  = 1 and for each b ∈ (X ∪ {∅}) , (X \{b})  = (Y \{a}), then there exists ∈ Da2 such that X  Y. (v) If a ∈ Y \ X , #Y  = 1 and Y  = (X ∪ {a}), then there exists ∈ Da1 such that X  Y.

552

C. Rodríguez-Álvarez

Part (i) says that if a candidate equipped with BDS1 preferences belongs to a set, then she always prefers this set to the set that contains all the candidates in the initial set but herself. Part (ii) says that if a candidate is equipped with BDS2 preferences and belongs to a set, then she always prefers that set to the set that is obtained by replacing herself with a new candidate. Parts (iii), (iv), and (v) imply that the preferences of the candidates over many sets of candidates are not restricted. Before proceeding with the analysis of BDS1-candidate stability, we introduce the following condition due to Arrow (1963) and proposed as a candidate stability condition by Eraslan and McLennan (2004). Strong candidate stability. For each a ∈ C and each P ∈ P r , either v(C , P) = {a} or v(C \{a}, P) = v(C , P)\{a}. Note that strong candidate stability is stronger than no-harm and insignificance. Besides no-harm and insignificance, strong candidate stability also incorporates the notion of no-expansion. If an elected candidate (who is not the only elected candidate) withdraws, then no additional candidate can be incorporated to the chosen set.6 With Lemma 1 at hand, we are in a position to analyze the implications of candidate stability in both BDS domains. The following lemma relates BDS1candidate stability and strong candidate stability. Lemma 2 Let C ∩ V = ∅. A voting correspondence v satisfies BDS1-candidate stability if and only if v satisfies strong candidate stability. Lemma 2 provides an expected utility rationale to Eraslan and McLennan’s (2004) strong candidate stability axiom. If candidates cannot vote, then strong candidate stability and BDS1-candidate stability are equivalent. The next lemma analyzes the implications of BDS2-candidate stability. Lemma 3 Let C ∩ V = ∅. A voting correspondence v satisfies BDS2-candidate stability if and only if (i) for each a ∈ C and each P ∈ P r such that a ∈ v(C , P) either (i.a) v(C , P) = {a}, or (i.b) if v(C , P)  = {a}, then either v(C , P)\{a} = v(C \{a}, P), or there is b ∈ C \{a}, such that v(C , P)\{a} = v(C \{a}, P)\{b} , and (ii) v satisfies insignificance. Not surprisingly, Lemma 3 shows that BDS2-candidate stability is weaker than BDS1-candidate stability. Note that (i) implies that if a voting correspondence satisfies BDS2-candidate stability then it satisfies no-harm. Moreover, part (i.b) implies that BDS2-candidate stability admits a minimal expansion in the chosen set after a candidate withdrawal. If an elected candidate withdraws, then she can be substituted in the chosen set by exactly one other candidate. At this point, we can check to which extent DJL’s main result applies to multivalued frameworks. First, we present different voting correspondences that extend the notion of dictatorships to the multivalued framework. 6 Formally, a voting correspondence v satisfies no-expansion if for each a ∈ C and each P ∈ P r , a ∈ v(C , P), and v(C , P)  = {a} imply v(C \{a}, P) ⊂ v(C , P).

Candidate stability and voting correspondences

553

A voting correspondence v is dictatorial if there is i ∈ V such that for each A ∈ A and each P ∈ P r , v (A, P) = top (A, Pi ).7 A voting correspondence v is bidictatorial if there are two distinct i, j ∈ V such that for each A ∈ A and each P ∈ P r , v (A, P) = top (A, Pi ) ∪ top(A, P j ). Let S ⊆ V . The voting correspondence v is the S-tops correspondence if for each A ∈ A and each P ∈ P r , v (A, P) = ∪i∈S top (A, Pi ). We say v is a tops correspondence if there is a non-empty set S ⊆ V , such that v is the S-tops correspondence. Let S ⊆ V . For each A ∈ 2C \{∅} and each PS ∈ P S , let Par (A, PS ) ≡ {a ∈ A, such that for no b ∈ A, b Pi a for each i ∈ S}. For each A ∈ A and each P ∈ P r , we say that b ∈ A is Pareto dominated at A if b∈ / Par(A, P). Let S ⊆ V . The voting correspondence v is the S-Pareto correspondence if for each A ∈ A and each P ∈ P r , v(A, P) = Par(A, PS ). Let S ⊆ V . A voting correspondence v is S-oligarchical if for each P ∈ P r , (i) v(C , P) ⊆ Par(C , PS ), (ii) For each a ∈ C , ∪i∈S top(C \{a}, Pi ) ⊆ v(C \{a}, P) ⊆ Par(C \{a}, PS ). (iii) For each a ∈ C , a ∈ ∪i∈S top(C , Pi ) and a ∈ / v(C , P) imply v(C , P) = C \{a}. We say that v is oligarchical if there is a non-empty set S ⊆ V such that v is S-oligarchical. Dictatorial, bidictatorial, tops, and S-Pareto correspondences are oligarchical. For every S ⊂ V such that #S = 1, every S-oligarchical voting correspondence is dictatorial. On the other hand, a bidictatorial voting correspondence is an S-oligarchical voting correspondence for some S ⊆ V with #S = 2. However, for each S ⊆ V with #S = 2 there are S-oligarchical voting correspondences that are not bidictatorial. It is clear that if C ∩ V = ∅, then every oligarchical voting correspondence is unanimous. Note also that under an oligarchical voting correspondence, the power of decision is concentrated in the hands of an arbitrary group of voters. This power of decision can be evenly shared by all the voters (S = V ), but at the cost of generating irresolute elections. In the following Theorems 1 and 2, we show that candidate stability and unanimity introduce severe restrictions in both BDS domains. If the candidates’ preferences over sets are conditional expected utility consistent, then almost every unanimous voting correspondence is subject to the candidates’ incentives to withdraw. Theorem 1 Let C ∩ V = ∅. A voting correspondence v satisfies BDS1-candidate stability and unanimity if and only if v is dictatorial. 7 This restricted definition of dictatorial voting correspondences is due to the fact that we only care about the stability of the whole set of candidates. The same remark applies to the following definitions of voting correspondences.

554

C. Rodríguez-Álvarez

Theorem 1 is Eraslan and McLennan’s (2004) Theorem for the case in which voters cannot run for office and are not allowed to express indifferences among candidates. Theorem 1 shows that DJL’s Theorem 1 can be extended to multivalued environments if candidates’ preferences over sets are restricted to be BDS1-consistent. A similar result holds for the BDS2 domain. Theorem 2 Let C ∩ V = ∅. (i) Let #C ≥ 4. Then, a voting correspondence v satisfies BDS2-candidate stability and unanimity if and only if v is either dictatorial or bidictatorial. (ii) Let #C = 3. Then, a voting correspondence v satisfies BDS2-candidate stability and unanimity if and only if v is oligarchical. In spite of reducing significatively the domain of admissible preferences over sets, Theorem 2 is in line with Theorem 1. Generically, only dictatorial and bidictatorial voting correspondences are admitted. However, when the initial set of potential candidates only contains three candidates, new possibilities arise. In that case, in addition to tops and S-Pareto correspondences, many voting correspondences satisfy BDS2-candidate stability and unanimity. Example 1 describes one such possibility. Example 1 Let C ∩ V = ∅. Assume #C = 3 and let V = {1, . . . , #V }. For each A ∈ 2C \{∅} and each P ∈ P r define the function k¯ : 2C \{∅} × P V → V in ¯ such a way that k(A, P) ≡ 1 if for each i ∈ V , top(A, Pi ) = top(A, P1 ) and ¯ k(A, P) ≡ min{i ∈ V such that top(A, Pi )  = top(A, P1 )} otherwise. Next, define v¯ as follows. For each A ∈ A and each P ∈ P r ,   . v¯ (A, P) ≡ top (A, P1 ) ∪ top A, Pk(A,P) ¯ Clearly, v¯ is V -oligarchical and satisfies BDS2-candidate stability. However, v¯ is neither dictatorial nor bidictatorial.

3.2 Candidate stability and leximin preferences We now assume that candidates are extreme risk averters. Hence, candidates compare sets of candidates using leximin preferences. Leximin preferences reflect the way in which sophisticated pessimistic candidates construct their preferences over sets.8 When comparing two sets, a candidate equipped with leximin preferences compares first the worst candidates in each set. If they are the same, then she compares the next to the worst ones, and if they are again the same, then she proceeds iteratively in the same fashion. Let a ∈ C , Pa ∈ P a and X ∈ 2C \{∅}. Let X 1 (Pa ) ≡ bottom(X, Pa ). Once t X t (Pa ) is defined for an integer t, let X t+1 (Pa ) ≡ {∅} if X = ∪i=1 X i (Pa ), and t X t+1 (Pa ) ≡ bottom([X \ ∪i=1 X i (Pa )], Pa ) otherwise. 8 Pattanaik (1973, 1978) was the first to use leximin preferences to analyze preferences over sets.

Candidate stability and voting correspondences

555

Let a ∈ C and Pa ∈ P a . The ordering ∈ D is the Pa -consistent leximin preference – ∈ Dalex (Pa ) – if and only if for each X, Y ∈ 2C \{∅}:



X  Y ⇔ X t (Pa )Pa Y t (Pa ), where t is the smallest integer t such that X t (Pa )  = Y t (Pa ). Note that for each Pa ∈ P a , there is only one Pa -consistent leximin preference over sets (#Dalex (Pa ) = 1). Moreover, the Pa -leximin preference over sets is Pa -BDS2 consistent. The following lemma shows that leximin preferences do not impose any no-expansion restriction over candidate stable voting correspondences. Lemma 4 Let C ∩ V = ∅. A voting correspondence v satisfies leximin-candidate stability stable if and only if v satisfies no-harm and insignificance. Lemma 4 provides a candidate stability rationale for no-harm and insignificance. It is not difficult to see that interesting voting correspondences satisfy the requirements of leximin-candidate stability. Proposition 1 Let C ∩ V = ∅. For each S ⊆ V , the S-tops correspondence and the S-Pareto correspondence satisfy leximin-candidate stability. Proof Let S ⊆ V . Consider two distinct a, b ∈ C , and P ∈ P r . Assume a, b ∈ ∪i∈S top(C , Pi ). Clearly, b ∈ ∪i∈S top(C \ {a}, Pi ), which proves that S-tops correspondence satisfies no-harm. Now assume a ∈ / ∪i∈S top(C , Pi ). Then, ∪i∈S top(C , Pi ) = ∪i∈S top(C \ {a}, Pi ), which proves that the S-tops correspondence satisfies insignificance. Finally, consider two distinct a, b ∈ C and P ∈ P r for which a, b ∈ Par (C , PS ). Note that if b is not dominated by any candidate in C , it is not dominated by any candidate in C \{a}. Then, b ∈ Par(C \{a}, PS ), which proves that the S-Pareto correspondence satisfies no-harm. Assume now that a ∈ / Par(C , PS ), but b ∈ Par(C , PS ). By the preceding argument b ∈ Par(C \{a}, PS ). Finally, assume that a ∈ / Par(C , PS ), and b ∈ / Par(C , PS ). Because a ∈ / Par(C , PS ), b is dominated by some candidate in C \{a}. Then b ∈ / Par(C \{a}, PS ), which proves that the S-Pareto correspondence satisfies insignificance.   The next theorem shows that although the leximin domain reduces the strategic incentives to withdraw, only oligarchical voting correspondences satisfy leximincandidate stability and unanimity. Theorem 3 Let C ∩ V = ∅. (i) If a voting correspondence v satisfies leximin-candidate stability and unanimity, then v is oligarchical. (ii) If #C = 3, then every oligarchical voting correspondence satisfies leximincandidate stability and unanimity. Theorem 3 provides a complete characterization of the family of leximin -candidate stable and unanimous voting correspondences when there are only three potential candidates and candidates cannot vote. Furthermore, this family of voting correspondences is identical to the family of BDS2-candidate stable and unanimity voting correspondences. However, for #C ≥ 4, we can find leximin-candidate stable and unanimous voting correspondences different from S-tops and S-Pareto correspondences.

556

C. Rodríguez-Álvarez

Example 2 Let C ∩ V = ∅, #C = 4, and V = {1, 2, 3, 4}. Let v be such that for each A ∈ 2C \{∅} and each P ∈ P r , v (A, P) ≡



C \{top(C , P4 )} 4 top (A, P ) ∪i=1 i

4 if ∪i=1 top(A, Pi ) = C , otherwise.

It is easy to check that v is leximin-candidate stable. Note that this example can be easily generalized to any society where #C ≤ #V .9 As we see in the following example, there are oligarchical voting correspondences that do not satisfy leximin-candidate stability. Example 3 Let C ∩ V = ∅ and assume that #V = 3. Let v be such that for each a ∈ C , each A ∈ 2C \{∅}, and each P ∈ P r , a ∈ v (A, P) if and only if a ∈ Par(A, P) and for some i ∈ V there are not two distinct b, c ∈ Par(A, P) such that b Pi a and c Pi a. Hence, v picks the candidates who are ranked first or second by some voter among the candidates who are not Pareto dominated. Note that v is V -oligarchical. When #C ≤ 5, v satisfies leximin candidate stability. However, when #C ≥ 6, v violates leximin candidate stability.10

4 Overlap between candidates and voters So far, we have analyzed the case in which candidates are not voters. However, in many settings of interest, candidates can also vote. We devote this section to analyze the implications of candidate stability in this relevant case. When there is a non-empty overlap between candidates and voters, candidate stability becomes less restrictive. Under the different domains of preferences we have analyzed, candidate stability no longer implies no-harm and insignificance. In fact, there exist non-oligarchical voting correspondences that satisfy the requirements of candidate stability and unanimity. When candidates are voters, the result of the election may depend on candidates’ preferences. By using the information contained in candidates’ preferences, we can devise punishments that may prevent the candidates from withdrawing. This fact is illustrated in the following example. Example 4 Let C ⊆ V . For each A ∈ 2C \{∅} and each P ∈ P r , define Par ∗ (A, P) ≡ {a ∈ A such that for no b ∈ A, b Pi a, for each i ∈ (V \{a})}. The extension of v to societies where #C > #V generates tops correspondences. It is clear that v satisfies insignificance. We now show that v satisfies no-harm. Let V = {1, 2, 3} and #C ≤ 5. Let a, b ∈ C and P ∈ P r . Assume, to the contrary, that a, b ∈ v (C , P), and b ∈ / v (C \{a}, P). Then, for some i ∈ V , there are two distinct c, d ∈ C \{a, b} such that c, d ∈ / Par(C , P), c, d ∈ Par(C \{b}, P), c Pi b, and d Pi b. As b ∈ Par(C , P), without loss of generality, there is j ∈ V , such that b P j a P j c P j d. Because #C ≤ 5, b ∈ v (C \{a}, P), and we obtain the desired contradiction. On the other hand, assume C = {a, b, c, d, e, f }. Let P ∈ P r be such that a P1 b P1 c P1 d P1 e P1 f , f P2 e P2 d P2 a P2 b P2 c, and e P3 f P3 d P3 a P3 c P3 b. It is easy to verify that Par(C , P) = {a, d, e, f } and Par(C \{a}, P) = {b, c, d, e, f }. Thus, v (C , P) = {a, d, e, f } and v (C \{a}, P) = {b, c, e, f }. Hence, if #C = 6, v violates no-harm and, by Lemma 4, leximin-candidate stability. 9

10

Candidate stability and voting correspondences

557

Let the voting correspondence v ∗ be such that for each a ∈ C and each P ∈ P r , v ∗ (C , P) ≡ Par ∗ (C , P), and v ∗ (C \{a}, P) ≡ bottom(Par ∗ (C \{a}, P), Pa ). Note that for each a ∈ C and each P ∈ P r , Par ∗ (C , P) ⊆ Par ∗ (C \{a}, P) ∪ {a} . Consider now two distinct a, b ∈ C and let P ∈ P r . Assume that v ∗ (C \{a}, P) = {b}. By the definition of v ∗ , there is no c ∈ v ∗ (C , P) such that b Pa c. Thus, for each ∈ Da1 (Pa ) ∪ Dalex (Pa ), v(C , P)  v(C \{a}, P) . Hence, v ∗ satisfies BDS1-candidate stability, BDS2-candidate stability, and leximin-candidate stability. Moreover, as candidates are also voters, by candidates’ self-preferences, there do not exist any unanimous profile of preferences. Hence, v ∗ trivially satisfies unanimity. However, v ∗ does not satisfy either no-harm or insignificance. Using punishments to eliminate candidates’ incentives to withdraw has a serious drawback. The result of the election after the withdrawal of a candidate may be detrimental to the interests of the remaining members of the society. In the light of Example 4, it is clear that no-harm and insignificance are not implied by candidate stability when some candidates are also voters. Nevertheless, these conditions are normatively appealing properties of voting correspondences. It is therefore of interest to determine which voting correspondences satisfy these properties when combined with a suitable unanimity condition. Because unanimity is vacuous when at least two candidates are voters, a stronger unanimity condition that ignores candidates’ self-preferences is employed instead. Let i ∈ V , A, B ∈ 2C \{∅}, and P ∈ P r . We say that B is a restricted top set at P relative to A if for each i ∈ V , each a ∈ A\(B ∪ {i}), and each b ∈ B, b Pi a. Strong unanimity. For each A ∈ A, each a ∈ A, and each P ∈ P r , if {a} is a restricted top set at P relative to A, then v (A, P) = {a}. Clearly, strong unanimity coincides with unanimity when C ∩ V = ∅. Theorem 4 Let C ∩ V  = ∅ (i) Let V \C  = ∅. If a voting correspondence v satisfies no-harm, insignificance, and strong unanimity, then there is a non-empty set of voters who are not candidates S ⊆ (V \C ) such that v is S-oligarchical. (ii) Let V ⊆ C . Then, there is no voting correspondence that satisfies no-harm, insignificance, and strong unanimity. Theorem 4 is our strongest result. In fact, the theorems in the previous section are corollaries to Theorem 4. When there are voters who are not candidates, allowing for some candidates to vote while strengthening candidate stability to noharm and insignificance does not generate new possibilities. The power of decision remains concentrated in the hands of an arbitrary group of voters who are not candidates. There is a clear explanation for this requirement. Let S ⊆ V be such that there is a ∈ S ∩ C . Let v be a S-oligarchical voting correspondence. By a’s selfpreference and by (ii) of the definition of an S-oligarchical voting correspondence, for each b ∈ C \{a} and each P ∈ P r , a ∈ v(C \{b}, P). But this would violate

558

C. Rodríguez-Álvarez

strong unanimity. The same fact also explains the impossibility result that holds when every voter is also a candidate.11

5 Conclusions In this article we have analyzed the extent to which the negative implications of candidate stability for single-valued voting procedures also apply to voting correspondences. First, we have studied settings where candidates cannot vote and they compare sets of candidates according to conditional expected utility maximization or according to leximin preferences. In these settings, we have proved that only oligarchical voting correspondences satisfy candidate stability and unanimity. In settings where candidates can also be voters, candidates’ preferences can be used to devise punishments that prevent candidates from withdrawing. Therefore, there do exist non-oligarchical voting correspondences that satisfy candidate stability and unanimity. Unfortunately, the selections proposed by these candidate stable voting correspondences may be detrimental to the interests of the other members of the society. A few final remarks are in order. It is clear that the implications of candidate stability depend on the domain of preferences over sets. Here, we have analyzed only three domains of preferences, but other domains are reasonable. For instance, candidates may be pessimistic, but not very sophisticated. When comparing two sets, they may only focus on the worst candidate in each set. Then, candidates use maximin preferences. Alternatively, candidates may be optimistic and sophisticated. When comparing two sets, they may focus (lexicographically) on the best candidate in each set. Then, candidates may use leximax preferences. Another possibility is that candidates are conditional expected utility maximizers, but they may be reluctant to express strict preference between two sets unless one clearly improves upon the other for every prior probability assessment. Then, candidates may use Gärdenfors preferences. In fact, when candidates cannot vote, maximincandidate stability is equivalent to the following condition. Whenever a candidate withdraws, all the candidates who were originally elected remain elected, but newly elected candidates can be introduced. On the other hand, leximax and Gärdenforscandidate stability are equivalent to insignificance. Clearly, in all those domains, candidate stability does not imply no-harm and insignificance. Thus, many voting correspondences satisfy candidate stability. However, in these domains candidate stability becomes so weak that no clear characterization is available.12 Finally, we devote a few words to clarify the relation between candidate stability and strategy-proofness. Candidate stability implies that a candidate never has an incentive to manipulate the result of a voting correspondence by withdrawing her candidacy. Strategy-proofness implies that, given a fixed agenda, a voter never has 11 When candidates are permitted to vote, DJL’s Theorem 3 shows that candidate stability is not compatible with a unanimity – (m − 2)-unanimity – condition that is consistent with a candidate’s self-preference and a weak monotonicity condition – top pair monotonicity. For the sake of brevity, we do not consider that result here. A similar result, available from the author, holds for voting correspondences that satisfy candidate stability for the domains of preferences analyzed here. 12 We refer the interested reader to a previous version of this work, Rodríguez-Álvarez (2003), for a more extensive discussion on this issue.

Candidate stability and voting correspondences

559

an incentive to manipulate the result of a voting correspondence by misrepresenting her preferences. Although related in spirit, both conditions are clearly independent. Candidate stability applies to variable agenda problems. Given a set of potential candidates, candidate stability relates the result of a voting correspondence at the agenda in which every candidate runs to the result when only one candidate withdraws. Strategy-proofness only applies to fixed agenda problems. Given a fixed agenda, strategy-proofness relates the result of a voting correspondence at different profiles of preferences in which only one voter changes her preferences. Despite the differences, the results when candidates are conditional expected utility maximizers and cannot vote are parallel to the results of Barberà et al. (2001) on strategy-proof social choice correspondences. However, when voters are assumed to have leximin preferences over sets, Campbell and Kelly (2000) have shown that, given a fixed agenda, only top correspondences satisfy strategy-proofness and unanimity. Thus, the similarities of the results depend crucially on the domain of preferences under consideration. 6 Appendices 6.1 Appendix 1: Proofs of the Theorems The proofs below do not appear in the same order as the results. We first prove our most general result, namely Theorem 4, and then use it to help prove the theorems in Sect. 3. The logic behind the proof of Theorem 4 runs as follows. We first analyze elections in which the voters’ preferences are restricted in such a way that three arbitrary candidates are preferred to the remaining candidates, and the voters’ preferences over the remaining candidates are fixed. In this restricted domain, noharm, insignificance, and strong unanimity imply that the voting correspondence is rationalizable by a group decision rule. Applying a version of Arrow’s (1963) Theorem by Mas-Colell and Sonnenschein (1972), we show that, on our restricted domain, the decision-power is concentrated in the hands of an arbitrary group of voters. Then, the joint implications of no-harm, insignificance, and strong unanimity allow us to extend this result to arbitrary profiles of preferences. Finally, strong unanimity implies that voters who are also candidates cannot be in the group of voters controlling the decision-power. This fact explains the impossibility result for societies in which every voter is also a candidate.13 Before proving Theorem 4, we introduce some further notation and definitions. Let B denote the set of all complete and reflexive binary relations on C . Analogously, for each A ∈ 2C \{∅}, B A denotes the set of all complete and reflexive binary relations on A. A binary relation Q ∈ B is quasitransitive if for each a, b, c ∈ C , a Q b and not b Q a, b Q c and not c Q b, imply a Q c and not c Q a. Hence, a binary relation is quasitransitive if its strict component is transitive. 13 The theorem admits other proofs. For instance, following Ehlers and Weymark (2003), we can show that if a voting correspondence satisfies strong unanimity, no-harm and insignificance, then it never selects Pareto dominated candidates at agendas containing at least #C −1 candidates. Then, applying the results of Denicolò (1987) on non-binary choice, we can complete the proof. Denicolò is not concerned with candidate stability, but he uses revealed preference axioms that are mathematically similar to no-harm and insignificance.

560

C. Rodríguez-Álvarez

A group decision rule is a mapping F : P r −→ B. Quasitransitivity. For each P ∈ P r , F (P) is quasitransitive. Arrow’s independence of irrelevant alternatives (AIIA). For each P, P ∈ P r , for each i ∈ V and each a, b ∈ C , P |{a,b} = P |{a,b} implies F(P) |{a,b} = F(P ) |{a,b} . Unanimity. For each a, b ∈ C , and each P ∈ P r , a Pi b for each i ∈ V , implies a F(P) b and not b F (P) a. Let S ⊆ V . The group decision rule F is S-oligarchical if for each a, b ∈ C and each P ∈ P r , a F (P) b if and only if there is i ∈ S with a Pi b. We say that the group decision rule F is dictatorial if there is i ∈ V such that F is {i}-oligarchical. The group decision rule F is oligarchical if there is a non-empty S ⊆ V such that F is S-oligarchical. Proposition 2 (Mas-Colell and Sonnenschein, 1972) Let F be a group decision rule defined on an unrestricted domain. If #V ≥ 2 and F satisfies quasitransitivity, AIIA, and unanimity, then F is oligarchical.14 Proof of Theorem 4 We establish the theorem from a sequence of lemmata. Henceforth, we assume that there exists a voting correspondence v that satisfies no-harm, insignificance, and strong unanimity. ¯ ⊆ P r denote the set of profiles Let A ∈ 2C \{∅} and let P¯ ∈ P r . Let P r (A, P) such that A is a restricted top set relative to C and P |C \A = P¯ |C \ A . Finally, let P r (A) ⊆ P r denote the set of profiles such that A is a restricted top set relative to C . Lemma 5 For each three distinct a, b, c ∈ C , each P¯ ∈ P r , and each P ∈  r P {a, b, c}, P¯ , v (C , P) ⊆ {a, b, c}.   Proof Let {a, b, c} ⊆ C and P¯ ∈ P r . Let P ∈ P r {a, b}, P¯ . By strong unanimity, v(C \{b}, P) = {a} . If b ∈ v(C , P), then by no-harm, v(C , P) ⊆ v(C \{b}, P) ∪ {b} = {a, b}. If b ∈ / v(C , P), then by insignificance, v( C , P) = v( C \{b}, P) = {a}. Thus, for each {a, b} ⊂ C and each P ∈ P r   ¯ {a, b}, P , v(C , P) ⊆ {a, b}. ¯ Let P ∈ P r ({a, b}, P) ¯ Finally, let d ∈ C \{a, b, c}. Let P ∈ P r ({a, b, c}, P). be such that P |C \{c} = P |C \{c} . By the arguments in the preceding paragraph, v(C , P ) ⊆ {a, b} . By insignificance, v(C , P ) = v(C \{c}, P ) . By (iii) of the definition of a voting correspondence, v(C \{c}, P ) = v(C \{c}, P ) . Thus, v(C \{c}, P ) ⊆ {a, b}. If c ∈ v(C , P ), then by no-harm, v(C , P ) ⊆ {a, b, c}. If c ∈ / v(C , P ), then by insignificance, v(C , P ) = v(C \{c}, P ) ⊆ {a, b} . Hence, d∈ / v(C , P ).   Next, for each three distinct a, b, c ∈ C define the auxiliary group decision rule,   F({a,b,c}, P¯ ) : P r {a, b, c}, P¯ −→ B{a,b,c} ,   in such a way that for each P ∈ P r {a, b, c}, P¯ , a F({a,b,c}, P¯ ) (P) b if and only if a ∈ v(C \{c}, P) . Note that by the previous Lemma, F({a,b,c}, P¯ ) is well-defined and, by construction, complete and reflexive. 14

As we assume that voters’ preferences are strict, the converse argument also holds.

Candidate stability and voting correspondences

561

Lemma 6 For each three distinct a, b, c ∈ C , and each P¯ ∈ P r , F({a,b,c}, P¯ ) is quasitransitive.   Proof Let {a, b, c} ⊆ C and P¯ ∈ P r . Let P ∈ P r {a, b, c}, P¯ be such v(C \{c}, P) = {a} and v(C \{a}, P) = {b}. Assume first that c ∈ v(C , P). By no-harm, v(C , P) ⊆ v(C \{c}, P) ∪ {c} = {a, c} . Assume now that c ∈ / v(C , P). Then, by insignificance, v(C , P) = v(C \{c}, P) = {a} . Hence, v(C , P) ⊆ {a, c}. Using the same arguments with candidate a, we obtain v(C , P) ⊆ {a, b}. Thus, v(C , P) = {a}. Finally, by insignificance, v(C , P) = v(C \{b}, P) = {a} , which proves that F({a,b,c}, P¯ ) is quasitransitive.   Lemma 7 For each three distinct a, b, c ∈ C , if V \{a, b, c}  = ∅, then for each P¯ ∈ P r there is a non-empty S ⊆ V \{a, b, c} such that F({a,b,c}, P¯ ) is S-oligarchical. Proof Let {a, b, c} ⊆ C be such that V \{a, b, c}  = ∅ and let P¯ ∈ P r . First, we ¯ if a Pi b for each i ∈ N \{a, b, c}, prove that for each P ∈ P r ({a, b, c}, P), ¯ then a F({a,b,c}, P¯ ) (P) b and not b F({a,b,c}, P¯ ) (P) a. Let P, P ∈ P r ({a, b, c}, P) be such that P |C \{c} = P |C \{c} , for each i ∈ N \{a, b, c}, a Pi b, for each i ∈ N \{b, c}, a Pi c Pi b, and b Pb a Pb c. By strong unanimity, v(C \{a}, P ) = {c}, and v(C \{b}, P ) = {a}. If a ∈ v(C , P ), then by no-harm, v(C , P ) ⊆ v(C \{a}, P ) ∪ {a} = {a, c}. If a ∈ / v(C , P ), then by insignifi cance, v(C , P ) = v(C \{a}, P ) = {c}. Hence, b ∈ / v(C , P ). By insignificance, v(C , P ) = v(C \{b}, P ) = {a}. Finally, by insignificance, v(C , P ) = v(C \{c}, P ) = {a}, and by (iii) of the definition of a voting correspondence v(C \{c}, P) = {a}. Thus, a F({a,b,c}, P¯ ) (P) b and not b F({a,b,c}, P¯ ) (P) a. ¯ Define the auxiliary group decision rule Let P{a,b,c} ∈ P {a,b,c} ({a, b, c}, P). ¯ → B{a,b,c} , FP : P N \{a,b,c} ({a, b, c}, P) ¯ in such a way that for each PN \{a,b,c} ∈ P N \{a,b,c} ({a, b, c}, P), FP (PN \{a,b,c} ) ≡ F({a,b,c}, P¯ ) (PN \{a,b,c} , P{a,b,c} ). Note that for each i ∈ N \{a, b, c}, preferences over candidates {a, b, c} are not restricted. By Lemma 6, FP is quasitransitive. By (iii) of the definition of a voting correspondence, FP satisfies AIIA. Moreover, by the arguments in the preceding paragraph, FP is unanimous. By (ii) of the definition of voting correspondence, preferences of candidates who are not voters do not affect FP . If #V \{a, b, c} = 1, then by FP ’s unanimity, FP is dictatorial. If #V \{a, b, c} > 1, then by Proposition 2, there is a non-empty set S ⊆ V \{a, b, c} such that FP is S-oligarchical. Finally, let P{a,b,c} ,P{a,b,c} ∈ P {a,b,c} ({a,b,c},P¯ ) be such that P|C \{c} =P |C \{c} . Hence, P{a,b,c} and P{a,b,c} only differ in candidates’ a and b preferences over candidate c. Define the auxiliary group decision rule FP in the same fashion we have defined FP . By the arguments in the preceding paragraph, there is a nonempty set S ⊆ V \{a, b, c} such that FP is S-oligarchical and a non-empty set S ⊆ V \{a, b, c} such that FP is S -oligarchical. Next, we show that S = S . Since we may pass from P{a,b,c} to any other restricted profile for {a, b, c} through a sequence of one-at-a-time changes, this suffices to prove the result.

562

C. Rodríguez-Álvarez

Assume, by way of contradiction, that there is an i ∈ S\S . Let  N \{a,b,c} ∈P {a, b, c}, P¯ be such that  a Pi b Pi c, andfor each j ∈ N \{a, b, c, i}, b P j a P j c. It is clear that P(N \{a,b,c}) , P{a,b,c} |C \{c} = P(N \{a,b,c}) , P{a,b,c} |C \{c} . Note that i ∈ S. Thus, we have that a ∈ v (C \{c},     / S , v C \{c}, P(N \{a,b,c}) , P(N \{a,b,c})  , P{a,b,c} . However, because i ∈ P{a,b,c} = {b}, which violates (iii) of the definition of a voting correspondence.   P (N \{a,b,c})

At this point, we introduce a family of voting correspondences. Let S ⊆ V and P¯ ⊆ P r . We say that v is S-oligarchical ∗ on the domain P¯ if for each A ∈ A and each P ∈ P¯ : (i) v(A, P) ⊆ Par(A, PS ). (ii) For each a ∈ A, if there exist i ∈ S and b ∈ A such that b Pi a, then v(A, P)  = {a}. Lemma 8 For each three distinct a, b, c ∈ C , if V \{a, b, c}  = ∅, then for each P¯ ∈ P r there is a non-empty S ⊆ V \{a, b, c} such that v is S-oligarchical ∗ on ¯ P r ({a, b, c}, P). Proof Let {a, b, c} ⊆ C be such that V \{a, b, c}  = ∅ and let P¯ ∈ P r . By Lemma 7, there is a non-empty set S ⊆ V such that F({a,b,c}, P) ¯ is S-oligarchical. Let ¯ P ∈ P r ({a, b, c}, P). We prove first part (i) of the definition of an S-oligarchical ∗ voting correspondence. Note that, by Lemma 5, v(C , P) ⊆ {a, b, c}. Assume that c ∈ / Par(C , PS ). Then, either a Pi c for each i ∈ S, or b Pi c for each i ∈ S. Without loss of generality, assume that for each i ∈ S, a Pi c. Thus, because F({a,b,c}, P) ¯ is S-oligarchical, v(C \{b}, P) = {a}. If b ∈ v(C , P), then by no-harm v(C , P) ⊆ v(C \{b}, P) ∪ {b} = {a, b}. If b ∈ / v(C , P), then by insignificance v(C , P) = v(C \{b}, P) = {a}. Thus, c ∈ / v(C , P), which suffices to prove that v(C , P) ⊆ Par(C , PS ). Next, we prove that for each d ∈ C , v(C \{d}, P) ⊆ Par(C \{d}, PS ). Note first that for each d ∈ C , Par(C , PS ) ⊆ Par(C \{d}, PS ) ∪ {d}.

(1)

Let d ∈ C . Assume first that d ∈ / v(C , P). By (i) of the definition of a voting correspondence, d ∈ / v(C \{d}, P). By insignificance, v(C , P) = v(C \{d}, P), and by the argument of the preceding paragraph v(C , P) ⊆ Par(C , PS ). Thus, (1) implies that v(C \{d}, P) ⊆ Par(C \{d}, PS ). ¯ Finally, assume that d ∈ v(C , P) (then d ∈ {a, b, c}). Let P ∈ P r ({a, b, c}, P) ¯ and P |C \{d} = P |C \{d} . Clearly, be such that P ∈ P r ({a, b, c}\{d}, P) Par(C \{d}, PS ) = Par(C \{d}, PS ). By the arguments in Lemma 5, d ∈ / v(C , P ). Hence, by the argument in the preceding paragraph, v(C \{d}, P ) ⊆ Par(C \{d}, PS ). By (iii) of the definition of a voting correspondence, v(C \{d}, P) = v(C \{d}, P ). Hence, we have that v(C \{d}, P) ⊆ Par(C \{d}, PS ). Next, we check part (ii) of the definition of an S-oligarchical ∗ voting cor¯ be such that for some i ∈ S, b Pi a. respondence. Let P ∈ P r ({a, b, c}, P)

Candidate stability and voting correspondences

563

Because F({a,b,c}, P) ¯ is S-oligarchical, by (iii) of the definition of an S-oligarchical group decision rule, b ∈ v(C \{c}, P). Assume to the contrary that v(C , P) = {a}. Then, by insignificance, v(C \{c}, P) = {a}, which contradicts that F({a,b,c}, P) ¯ is S-oligarchical. Thus, v(C , P)  = {a}. Note that for each d ∈ C \{a, b, c}, by Lemma 5, d ∈ / v(C , P). By insignificance, v(C , P) = v(C \{d}, P). Hence, for each d ∈ C \{a, b, c}, v(C \{d}, P)  = {a}.     r r Note that, if C = {a, b, c}, then P = P {a, b, c}, P¯ . We extend the preceding lemma to arbitrary sizes of the set of candidates using an induction argument. We introduce first a pair of lemmata. Lemma 9 Let #C ≥ 4. For each three distinct a, b, c ∈ C , if V \{a, b, c}  = ∅, then there is a non-empty set S ⊆ V \{a, b, c} such that v is S-oligarchical ∗ on P r ({a, b, c}). ˆ P˜ ∈ P r ({a, b, c}) be such that there is d ∈ Proof Let {a, b, c} ⊂ C . Let P, ˆ ˜ C \{a, b, c} for which Pˆ |C \{d} = P˜ |C \{d} . By Lemma 8,  there are S, S ⊆ ∗ r ˆ ˆ V \{a, b, c} such that v is S-oligarchical on P {a, b, c}, P , and   ∗ on P r {a, b, c}, P˜ . To establish the Lemma, as P, ˆ P, ˜ and d ˜ S-oligarchical are arbitrary and the argument can be applied iteratively, we need only show ˆ ˜ that Sˆ = S˜ . Hence, assume,  by way of contradiction, that there is i ∈ S\ S. r r Let P ∈ P {a, b, c}, Pˆ and P ∈ P {a, b, c}, P˜ be such that P |C \{d} =

P |C \{d} , a Pi b Pi c, and for  each j ∈ V \{i}, b P j a P j c. Then, a ∈ v (C \{d}, P), while v C \{d}, P = {b}, which violates (iii) of the definition of a voting correspondence.  

Lemma 10 Let #C ≥ 4. If V \C  = ∅, then there is a non-empty set S ∈ V \C such that for every three distinct a, b, c ∈ C , v is S-oligarchical ∗ on P r ({a, b, c}). Proof It is enough to consider any two distinct {a, b, c} and {a, b, d} and to show that there is a set S ⊆ V \{a, b, c, d} such that v is S-oligarchical ∗ on P r ({a, b, c}) and P r ({a, b, d}). As the choice of c and d is arbitrary and we can repeat the argument iteratively, this suffices to prove the result. Consider two distinct set of candidates {a, b, c} and {a, b, d}. By the preceding lemma, there are S ⊆ V \{a, b, c}, and S ⊆ V \{a, b, d}, such that v is S-oligarchical ∗ on P r ({a, b, c}), while v is S -oligarchical ∗ on P r ({a, b, d}). Assume, by way of contradiction, that there exists i ∈ S\S . Let P¯ ∈ P r ({a, b}) ∩ P r ({a, b, c, d}) be such that a Pi b, while for each j ∈ V \{b, i}, b P j a. Let P ∈ P r ({a, b, c}) be such that P |C \{d} = P¯ |C \{d} . As v is S-oligarchical ∗ on P r ({a, b, c}), a ∈ v(C \{c}, P). Next, let P ∈ P r ({a, b, d} be such that P |C \{c} = P |C \{c} . Because v is S -oligarchical ∗ on P r ({a, b, d}) and i ∈ / S , v(C \{c}, P ) = {b}; which violates (iii) of the definition of a voting correspondence.   Lemma 11 Let C ≥ 4. If V \C  = ∅, then there is a non-empty set S ⊆ V \C such that v is S-oligarchical ∗ on P r . Proof Using an induction argument, we extend Lemma 10 to arbitrary profiles of preferences.

564

C. Rodríguez-Álvarez

Induction Step For some integer m ≥ 3, there is S ⊆ V \C such that for each B ∈ 2C \{∅} with # B = m, v is S-oligarchical ∗ on P r (B): Now we show that the result is true for # B = (m + 1). Let B ∈ 2C \{∅} be such that # B = (m + 1). Let P ∈ P r (B) . Assume, contrary to part (i) of the definition of an S- oligarchical ∗ voting correspondence, that there is a ∈ v(C , P) such that a ∈ / Par(C , PS ). Then, there is b ∈ B such that for each i ∈ S, b Pi a. Let c ∈ B\{a, b} (such a candidate exists because # B ≥ 3). Clearly, a ∈ / Par(C \ {c}, PS ). Let P ∈ P r (B\{c}) be such that  P |C \{c}= P |C\{c} . Note that  # (B\{c}) = m. By the induction hypothesis, v C \{c}, P ⊆ Par C \{c}, PS . Because Par(C \{c}, PS ) = Par(C \{c}, PS ) and a∈ / Par(C \{c}, PS ), this implies that a ∈ / v(C\{c}, P ).By (iii) of the definition of a voting correspondence, v (C \{c}, P) = v C \{c}, P . Thus, a ∈ / v(C \{c}, P). Assume now that c ∈ v(C , P). Then by no-harm, v (C , P) ⊆ v (C \{c}, P) ∪ {c}, and, hence, a ∈ / v(C , P). Finally, assume that c ∈ / v(C , P). Then by insignificance, v(C , P) = v(C \{c}, P), and, hence, a ∈ / v(C , P). Therefore, v(C , P) ⊆ Par(C , PS ). Next, let d ∈ C and let P ∈ P r (B). We prove that v(C \{d}, P) ⊆ Par(C \{d}, PS ). Assume that d ∈ v(C , P) (thus, d ∈ B). Note that #(B\{d}) = m. Let P ∈ P (B\{d}) be such that P |C \{d} = P |C \{d} . By the induction hypothesis v(C \{d}, P ) ⊆ Par(C \{d}, PS ). By (iii) of the definition of a voting correspondence, v(C \{d}, P) = v(C \{d}, P ). Then, v(C \{d}, P) ⊆ Par(C \{d}, PS ) because Par(C \{d}, PS ) = Par(C \{d}, PS ). Finally, assume that d ∈ / v(C , P). By the argument of the preceding paragraph, v(C , P) ⊆ Par(C , PS ). Note also that Par(C , PS ) ⊆ Par(C \{d}, PS ) ∪ {d} . By insignificance, v(C , P) = v(C \{d}, P). Hence, v(C \{d}, P) ⊆ Par(C \{d}, PS ), and part (i) of the definition of an S-oligarchical ∗ voting correspondence is satisfied when # B = m + 1. We now check part (ii) of the definition of an S- oligarchical ∗ voting correspondence. Let a, b ∈ C and i ∈ S. Let P ∈ P r be such that b Pi a. Assume, by way of contradiction, that v(C , P) = {a}. Let c ∈ B\{a, b}. By insignificance, v(C \{c}, P) = {a}. Let P ∈ P r (B\{c}) be such that P |C \{c} = P |C \{c} . Note that #(B\{c}) = m. Then by the induction hypothesis, v(C \{c}, P )  = {a}, which contradicts (iii) of the definition of a voting correspondence. Thus, v(C , P)  = {a}. Finally, we check that for each c ∈ / {a, b}, v(C \{c}, P)  = {a}. Let c ∈ C \{a, b}. Assume c ∈ v(C , P) (then, by part (i) of the definition of an S-oligarchical ∗ voting correspondence, c ∈ B). Let P ∈ P r (B\{c}) be such that P |C \{c} = P |C \{c} . By the induction hypothesis, v(C \{c}, P )  = {a}. By (iii) of the definition of a voting correspondence, v(C \{c}, P) = v(C \{c}, P ). Thus, v(C \{c}, P)  = {a}. Finally, assume c ∈ / v(C , P). By the preceding argument, v(C , P)  = {a}. By insignificance, v(C , P) = v(C \{c}, P). Thus, for each c ∈ C \{a, b}, v(C \{c}, P)  = {a}, and part (ii) of the definition of an S-oligarchical ∗ voting correspondence is satisfied when # B = m + 1.   Now, we are in a position to conclude the proof of Theorem 4. Proof of Theorem 4.(i) Assume that V \C  = ∅. By Lemma 11, there is a nonempty S ⊆ V \C such that v is S-oligarchical ∗ . It is immediate that v satisfies part (i) and the second set inclusion of part (ii) of the definition of an S-oligarchical voting correspondence.

Candidate stability and voting correspondences

565

Next, consider two distinct a, b ∈ C and i ∈ S. Let P ∈ P r be such that b = top(C \{a}, Pi ). Let P ∈ P r be such that b = top(C , Pi ), a = top(C \{b}, Pi ), while a = top(C , P j ) for each j ∈ S, and P |C \{a} = P |C \{a} . Because v is S-oligarchical∗ , we have that v(C , P ) = {a, b}. By no-harm, b ∈ v(C \{a}, P ). Then, by (iii) of the definition of a voting correspondence, b ∈ v(C \{a}, P), which proves part (ii) of the definition of an S-oligarchical voting correspondence. Finally, assume, by way of contradiction, that there are a ∈ C and P ∈ P r such that a ∈ ∪i∈S top(C , Pi ), a ∈ / v(C , P), and v(C , P)  = C \{a}. Then, there is b ∈ C \{a} such that b ∈ / v(C , P). By part (ii) of the definition of an S-oligarchical voting correspondence, a ∈ v(C \{b}, P) . However, by insignificance, v(C , P) = v(C \{b}, P). Thus, a ∈ / v(C \{b}, P) , which is a contradiction and proves part (iii) of the definition of an S-oligarchical voting correspondence.   Proof of Theorem 4.(ii) Assume, by way of contradiction, that V ⊆ C and that there is a voting correspondence v that satisfies no-harm, insignificance, and strong unanimity. Let P¯ ∈ P r . Suppose first that there are two distinct a, b ∈ C such that ¯ Note that a Pa b and b Pb a. Hence, both V = {a, b}. Let P ∈ P r ({a, b}, P). {a} and {b} are restricted top sets at P relative to C . Thus, by strong unanimity, v(C , P) = {a} and v(C \{c}, P) = {b}, a contradiction. Next, suppose that there are three distinct a, b, c ∈ C such that V = {a, b, c}. ¯ be such that a Pa b Pa c, b P c P a, and c Pc a Pc b. Let P ∈ P r ({a, b, c}, P) b b By strong unanimity, v(C \{a}, P ) = {b}, v(C \{b}, P ) = {c}, and v(C \{c}, P ) = {a}, which is a contradiction, because by Lemma 6, F({a,b,c}, P) ¯ is quasitransitive. Finally, suppose that #V ≥ 4. By Lemmata 8 and 9, for each three distinct a, b, c ∈ C , there is a non-empty S ⊆ V \{a, b, c} such that v is S-oligarchical ∗ on P ({a, b, c}). The same arguments we employed in Lemma 10 apply to prove that there is a non-empty set S ⊆ V \C such that for each {a, b, c} ⊂ C , v is Soligarchical ∗ on P ({a, b, c}), which is a contradiction because V \C = ∅.   Proof of Theorem 1 It is immediate that dictatorial voting correspondences satisfy BDS1-candidate stability and unanimity. Hence, we focus on the converse statement. Let v satisfy BDS1-candidate stability and unanimity. Because C ∩ V = ∅, by Lemma 2, v satisfies no-harm and insignificance. Moreover, v satisfies strong unanimity. Then, by Theorem 4, there is a non-empty S ⊆ V such that v is S-oligarchical. Assume, contrary to Theorem 1, that there are two distinct i, j ∈ S. Consider three distinct a, b, c ∈ C . Let P ∈ P r ({a, b, c}) be such that a Pi b Pi c, while for each j ∈ V \{i}, b P j c P j a. Then, v (C , P) = {a, b} and v (C \{b}, P) = {a, c}, which, by Lemma 2, violates BDS1-candidate stability.   Proof of Theorem 2 We first prove sufficiency of part (i) and part (ii) of Theorem 2. It is clear that dictatorial and bidictatorial voting correspondences satisfy BDS2candidate stability and unanimity. Moreover, every oligarchical voting correspondence satisfies unanimity. Hence, we only have to check that if #C = 3, then every oligarchical voting correspondence satisfies BDS2-candidate stability. Let C = {a, b, c} and let S ⊆ V be non-empty. Let v be an S-oligarchical voting correspondence. Let P ∈ P r . We have three possibilities. First, assume, that v(C , P) = {a, b, c}. Then, because v is S-oligarchical, Par(C , PS ) = {a, b, c}, Moreover, Par(C \{c}, PS ) = {a, b}. Then, by the definition of an S-oligarchical voting correspondence, v(C \{c}, P) = {a, b}. Now assume that v(C , P) = {a, b}.

566

C. Rodríguez-Álvarez

Thus, {a, b} ⊆ Par(C , PS ) and a ∈ Par(C \{b}, P). Hence, we have that either v(C \{b}, P) = {a} or v(C \{b}, P) = {a, c}. Note also that, because Par(C , PS ) ⊆ Par(C \{c}, PS ) ∪ {c}, then we have Par(C \{c}, PS ) = {a, b}. Thus, because v is S-oligarchical, v(C \{c}, P) = {a, b}. Finally, assume that v(C , P) = {a}. Then, because v is S-oligarchical, Par(C , PS ) = {a}. Moreover, for each b ∈ C \{a} , Par(C \{b}, PS ) = {a} , and by the definition of an S-oligarchical voting correspondence, v(C \{b}, P) = {a} . It is immediate to check that for each of these three possibilities, v satisfies conditions (i) and (ii) of Lemma 3 at profile P. The three cases exhaust all the possibilities up to a relabeling of the candidates. Hence, every oligarchical voting correspondence satisfies BDS2-candidate stability when #C = 3. Next, we prove necessity of part (i) and part (ii) of Theorem 2. Let v satisfy BDS2-candidate stability and unanimity. By Lemma 3, v satisfies no-harm and insignificance. Moreover, v satisfies strong unanimity. Thus, by Theorem 4, there is a non-empty S ⊆ V such that v is S-oligarchical. This suffices to prove necessity of part (ii) of Theorem 2. From now on, assume that #C ≥ 4. First, we show that #S ≤ 2. Assume, contrary to Theorem 2, that there are three distinct i, j, k ∈ S. Consider four distinct a, b, c, d ∈ C . Let P ∈ P r ({a, b, c}) be such that a = top(C , Pi ), b = top(C , P j ), and c = top(C , Pk ) for each k ∈ V \{i, j}. Because v is S-oligarchical, v(C \{d}, P) = {a, b, c}. Let P ∈ P r be such that a = top(C , Pi ), d = top(C , P j ) for each j ∈ V \{i}, and P |C \{d} = P |C \{d} . Note that Par(C , PS ) = {a, d}. Then, because v is S-oligarchical, v(C , P ) = {a, d}. However, by (iii) of the definition of a voting correspondence, v(C \{d}, P ) = {a, b, c}, which, by Lemma 3.(i), violates BDS2-candidate stability. Thus, #S ≤ 2. It remains to check that v is either dictatorial or bidictatorial. If #S = 1, then v is indeed dictatorial. Hence, assume that there are two distinct i, j ∈ V such that S = {i, j}. First, it is shown that for each P ∈ P r , v(C , P) = top(C , Pi ) ∪ top(C , P j ). Consider four distinct a, b, c, d ∈ C . Let P ∈ P r ({a, b, c, d}) be such that a = top(C , Pi ) and b = top(C , P j ) . On the contrary, assume that c ∈ v(C , P) . By no-harm and insignificance, c ∈ v(C \{d}, P) . Let P ∈ P V be such that a = top(C , Pi ), d = top(C \{a}, Pi ), d = top(C , P j ), and P |C \{d} = P |C \{d} . Because v is S-oligarchical, v(C , P ) = {a, d}. By part (ii) of the definition of an S-oligarchical voting correspondence, {a, b} ⊆ v(C \{d}, P ) . Moreover, by (iii) of the definition of a voting correspondence, v(C \{d}, P) = v(C \{d}, P ) . Hence, {a, b, c} ⊆ v(C \{d}, P ) , which, by Lemma 3.(i), violates BDS2-candidate stability. Hence, v(C , P) ⊆ top(C , Pi ) ∪ top(C , P j ). Finally, note that #C ≥ 4 and #S = 2. Then, by (iii) of the definition of an S-oligarchical voting correspondence, top(C , Pi ) ∪ top(C , P j ) ⊆ v(C , P) . Thus, v(C , P) = top(C , Pi ) ∪ top(C , P j ) . In order to conclude the proof, let c ∈ C and let P ∈ P r . Let P ∈ P r be such that P |C \{c} = P |C \{c} and c = bottom(C , Pi ) for each i ∈ V . Clearly, top(C , Pi ) ∪ top(C , P j ) = top(C \{c}, Pi ) ∪ top(C \{c}, P j ). By the preceding argument, v(C , P ) = top(C , Pi )∪top(C , P j ). By insignificance, v(C , P ) = v(C \{c}, P ) . By (iii) of the definition of a voting correspondence, v(C \{c}, P) = v(C \{c}, P ). Thus, v(C \{c}, P) = top(C \{c}, Pi )∪top(C \{c}, P j ).  

Candidate stability and voting correspondences

567

Proof of Theorem 3 Part (i) follows directly from Theorem 4 and Lemma 4. Finally, note that because C ∩ V = {∅}, it is clear from Lemmata 3 and 4 that BDS2-candidate stability implies leximin-candidate stability. Thus, part (ii) follows from part (ii) of Theorem 2.   6.2 Appendix 2: Proofs of the Lemmata of Sect. 3 Proof of Lemma 1: Proof of (i). Assume, to the contrary, that there is ∈ Da1 such that Y  X . Then, there are Pa ∈ P a , u a representing Pa , and a prior probability assessment λ such that:  λ (y) u a (y)  λ (y) u a (y) λ (a) u a (a)   ≥ .  +  λ (a) + y ∈Y λ (y ) λ (a) + λ (y ) λ (y ) y∈Y

y ∈Y

y ∈Y

y∈Y

(2) Note that the RHS of (2) is a convex combination of u a (a) and the LHS of (2) with positive weights. Because u a (a) > u a (y) for each y ∈ Y and because λ(a) > 0, the RHS of (2) exceeds the LHS of (2), a contradiction. Proof of (ii). Let  be a Pa -consistent BDS2 preference for some Pa ∈ P a . Note that a Pa b, # X = #Y , and (X \{a}) = (Y \{b}) = (X ∩ Y ) . Thus, for each u a representing Pa , u a (a) > u a (b), and



  1 1 u a (x) > u a (x) . u a (a) + u a (b) + #X #Y x∈X ∩Y

x∈X ∩Y

Hence, X  Y . Proof of (iii). Because a ∈ / (X ∪ Y ), then there is Pa ∈ P a such that for each b ∈ (X \Y ), each c ∈ (X ∩ Y ), and each d ∈ (Y \X ), b Pa c Pa d. Moreover, there is a utility function u a representing Pa such that for each b ∈ (X \Y ), each c ∈ (X ∩ Y ), and each d ∈ (Y \X ), u a (b) > u a (c) > u a (d). Then, we get 1  1  u a (x) > u a (y) . #X #Y x∈X

y∈Y

Hence, there exists a ∈ Da2 such that X  Y . Proof of (iv). We have two cases: Case (iv-a): There is c ∈ C \{a}, c ∈ Y \X . Let Pa ∈ P a be such that c = bottom (C , Pa ) . Let u a represent Pa and be such that ⎛ ⎞   #Y u a (c) < ⎝ u a (x) − u a (y)⎠ . #X x∈X

Rearranging terms, we get

1 #X

 x∈X

u a (x) >

y∈Y

1 #Y



y∈Y \{c} u a (y).

568

C. Rodríguez-Álvarez

Case (iv-b): (Y \{a}) ⊂ X . In this case, the hypothesis of (iv) implies that there exist two distinct b, b ∈ X \Y . Let B ≡ X \Y . Let Pa ∈ P a be such that for each b ∈ B and each c ∈ C \(B ∪ {a}), b Pa c. For each ε ∈ (0, 1), we can define a utility function u aε representing Pa in such a way that, u aε (a) = 1, for each b ∈ B, u aε (b) > (1 − ε), and for each c ∈ C \(B ∪ {a}), u aε (c) < 0. Note that for each ε ∈ (0, 1),

 1  ε 1 u a (x) > #Y +# B−1 (1 − ε)# B + u aε (y ) , and #X ∈Y \{a} x∈X y

 1  ε 1 ε u a (y) = #Y 1 + u a (y ) . #Y y ∈Y \{a}

y∈Y

Because # B ≥ 2,



ε y ∈Y \{a} u a (y )

#Y + # B − 1

 >

ε y ∈Y \{a} u a (y )

#Y

.

(#Y −1)(# B−1) , #Y # B

Moreover, because #Y ≥ 2 and # B ≥ 2, for each ε <

1 (1 − ε )# B > . #Y + # B − 1 #Y B−1) 1  , # X x∈X u aε (x) > Hence, for each ε < (#Y −1)(# #Y # B

1 #Y





u aε (y).

y∈Y

Proof of (v). Because D2 ⊂ D1 , by Lemma 1.(iv), we only have to check the following case. Let X, Y be such that for some b ∈ X \Y , X \{b} = Y \{a}. Let Pa ∈ P a be such that b = top(C \{a}, Pa ). For each ε ∈ (0, 1) define the utility function representing Pa , u aε in such a way that u aε (a) = 1, u aε (b) = (1 − ε), while for each c ∈ C \{a, b}, u aε (c) < 0. Next, define the probability assessment λ in such a way that λ(b) = 21 , and for each c ∈ C \{b}, λ(c) = 2(#C1−1) . Note that ⎞ ⎛  ε (x)  λ(x)u 1 1 a x∈X ⎝ (1 − ε) +  u aε (y )⎠ , and = 1 #Y −1 λ(x) 2 2(# C − 1) + x∈X 2 2(#C −1) y ∈Y \{a} ⎞ ⎛  ε (y)  λ(y)u 1 1 1 a y∈Y  u aε (y )⎠ . = #Y ⎝ + λ(y) 2(# C − 1) 2(# C − 1) y∈Y 2(#C −1) y ∈Y \{a}

Clearly, because #C ≥ 3, 

ε y ∈Y \{a} u a (y )

(#C − 1) + (#Y − 1) while for each ε <

 >

ε y ∈Y \{a} u a (y )

#Y

,

(#Y −1)(#C −2) #Y (#C −1) ,

(1 − ε )(#C − 1) 1 > . (#C − 1) + (#Y − 1) #Y Hence, for each ε <

(#Y −1)(#C −2) #Y (#C −1) ,



we get



λ(x)u aε (x) x∈X λ(x)

x∈X 



>



λ(y)u aε (y) y∈Y λ(y)

y∈Y 

.

 

Candidate stability and voting correspondences

569

Proof of Lemma 2 Assume that v satisfies strong candidate stability. Let a ∈ C and P ∈ P r . There are three cases. First, assume that v(C , P) = {a}. Then, by candidates’ self-preference, for each ∈ Da1 (Pa ), v(C , P)  v(C \{a}, P). Second, assume that a ∈ v(C , P), but v(C , P)  = {a}. Then, we have that v(C , P)\{a} = v(C \{a}, P). Hence, by Lemma 1.(i), for each ∈ Da1 (Pa ), v(C , P)  v(C \{a}, P). Finally, assume a ∈ / v(C , P). By insignificance, v(C , P) = v(C \{a}, P), and clearly, for each ∈ Da1 (Pa ), v(C , P)  v(C \{a}, P). Assume now that v satisfies BDS1-candidate stability. Assume, to the contrary, that v does not satisfy strong candidate stability. Suppose first that there are a ∈ C and P ∈ P r such that a ∈ v(C , P), v(C , P)  = {a}, but v(C \{a}, P)  = v(C , P)\{a}. By Lemma 1.(v), for some Pa ∈ P a there is  ∈ Da1 (Pa ) such that v(C \{a}, P)  v(C , P). Let P¯ ∈ P r be such that Pa = P¯a , while for each i ∈ V , Pi = P¯i . By (ii) of the definition of voting correspondence, v(C , P) = ¯ and v(C \{a}, P) = v(C \{a}, P), ¯ which violates BDS1-candidate stability. v(C , P) Finally, suppose that for some a ∈ C and P ∈ P r , a ∈ / v(C , P) but v(C , P)  = v(C \{a}, P). By Lemma 1.(iii), for some Pa ∈ P a there is  ∈ Da1 (Pa ) such that v(C \{a}, P)  v(C , P). Reasoning as above, this violates BDS1-candidate stability.   Proof of Lemma 3 Assume that v satisfies parts (i) and (ii) of Lemma 3. Note that for each a ∈ C , Da2 ⊂ Da1 . Then, by the arguments in the proof of Lemma 2, we only have to check the following possibility. Let a, b ∈ C , P ∈ P r be such that a ∈ v(C , P), v(C , P)  = {a} , and v(C , P)\{a} = v(C \{a}, P)\{b} . By Lemma 1.(ii), for each ∈ Da2 (Pa ), v(C , P)  v(C \{a}, P) . Thus, v satisfies BDS2-candidate stability. Let v satisfy BDS2-candidate stability. Assume, to the contrary, that v does not satisfy part (i) of Lemma 3. Then, there are a ∈ C , P ∈ P r , such that a ∈ v(C , P), v(C , P)  = {a} , but neither v(C , P) = v(C {a}, P) ∪ {a} , nor there is b ∈ C \{a} such that v(C , P)\{a} = v(C \{a}, P)\{b} . Then, by Lemma 1.(iv), for some Pa ∈ P a there is  ∈ Da2 (Pa ) such that v(C \{a}, P)  v(C , P). Let Pˆ ∈ P r be such that Pa = Pˆa , while for each i ∈ V , Pi = Pˆi . By (ii) of the definition of voting ˆ and v(C \{a}, P) = v(C \{a}, P), ˆ which viocorrespondence, v(C , P) = v(C , P) lates BDS2-candidate stability. The arguments of the proof of Lemma 2 apply to prove that v satisfies insignificance.   Proof of Lemma 4 Assume that v satisfies no-harm and insignificance. Let a ∈ C . Let P ∈ P r be such that a ∈ v(C , P). First, suppose that {a} = v(C , P) . By candidates’ self-preference, for the unique ∈ Dalex (Pa ), v(C , P)  v(C \{a}, P). Second, suppose that a ∈ v(C , P), {a}  = v(C , P). For the sake of notational simplicity, let X ≡ v(C , P), and Y ≡ v(C \{a}, P). Let t be the smallest integer such that X t (Pa )  = Y t (Pa ). We have two possibilities. Either X t (Pa ) ∈ Y or / Y . Assume first that X t (Pa ) ∈ Y . By no-harm, there is an integer X t (Pa ) ∈ t > t such that X t (Pa ) = Y t (Pa ). Then, X t (Pa ) Pa Y t (Pa ), and for the / Y. unique ∈ Dalex (Pa ), v(C , P)  v(C \{a}, P). Assume now that X t (Pa ) ∈ t t Note that X (Pa )  = ∅ because a ∈ X \Y . Then, by no-harm, X (Pa ) = a. By candidates’ self-preference, X t (Pa ) Pa Y t (Pa ). Hence, v(C , P)  v(C \{a}, P) . Finally, suppose that a ∈ / v(C , P). By insignificance, v(C , P) = v(C \{a}, P), and, hence, for the unique ∈ Dalex (Pa ), v(C , P)  v(C \{a}, P) .

570

C. Rodríguez-Álvarez

Now, let v satisfy leximin-candidate stability. Assume, contrary to Lemma 4, that v does not satisfy no-harm. Then, there are two distinct a, b ∈ C , and P ∈ P r such that a, b ∈ v(C , P), b ∈ / v(C \{a}, P). Let Pa ∈ P a be such that lex b = bottom(C , Pa ). Then, for  ∈ Da (Pa ), v(C \{a}, P)  v(C , P). Let P˜ ∈ P r be such that Pa = P˜a , while for each i ∈ V , Pi = P˜i . By (ii) of the definition ˜ and v(C \{a}, P) = v(C \{a}, P) ˜ , of voting correspondence, v(C , P) = v(C , P) which violates leximin-candidate stability. Next, assume, contrary to Lemma 4, that v does not satisfy insignificance. Then, there are a ∈ C and P ∈ P r such that a ∈ / v(C , P), and v(C , P)  = v(C \{a}, P). Let Pa ∈ P a be such that for each b ∈ v(C \{a}, P)\v(C , P), each c ∈ v(C \{a}, P) ∩ v(C , P), and each d ∈ v(C , P)\v(C \{a}, P), b Pa c Pa d. Then, for the unique  ∈ Dalex (Pa ), v(C \{a}, P)  v(C , P), which, by (ii) of the definition of voting correspondence and an already familiar argument, violates leximin-candidate stability.   References 1. Arrow KJ (1963) Social choice and individual values. 2nd edn. Wiley, New York 2. Barberà S, Bossert W, Pattanaik PK (2004) Ranking sets of objects. In: Barberà S, Hammond PJ, Seidl C (eds) Handbook of utility theory, vol 2. Kluwer, Boston, pp 893–977 3. Barberà S, Dutta B, Sen A (2001) Strategy-proof social choice correspondences. J Econ Theory 101:374–394 4. Berga D, Bergantiños G, Massó J, Neme A (2004) Stability and voting by committees with exit. Soc Choice Welfare 23:229–247 5. Besley T, Coate S (1997) An economic model of representative democracy. Q J Econ 112:85– 114 6. Campbell D, Kelly JS (2000) A trade-off result for preference revelation. J Math Econ 34:129–141 7. Denicolò V (1987) Some further results on non-binary social choice. Soc Choice Welfare 4:277–285 8. Dutta B, Jackson MO, Le Breton M (2001) Strategic candidacy and voting procedures. Econometrica 69:1013–1038 9. Dutta B, Jackson MO, Le Breton M (2002) Voting by successive elimination and strategic candidacy. J Econ Theory 103:190–218 10. Ehlers L, Weymark J (2003) Candidate stability and non-binary choice. Econ Theory 22:233– 243 11. Eraslan H, McLennan A (2004) Strategic candidacy and multivalued voting procedures. J Econ Theory 117:29–54 12. Mas-Colell A, Sonnenschein H (1972) General possibility theorems for group decisions. Rev Econ Stud 39:185–192 13. Osborne MJ, Slivinski A (1996) A model of political competition with citizen candidates. Q J Econ 111:65–96 14. Pattanaik PK (1973) On the stability of sincere voting situations. J Econ Theory 6:558–574 15. Pattanaik PK (1978) Strategy and group choice. North-Holland, Amsterdam 16. Rodríguez-Álvarez C (2003) Candidate stability and voting correspondences. Warwick Economics Research Papers No. 666, University of Warwick 17. Rodríguez-Álvarez C (2006) Candidate stability and probabilistic voting procedures. Econ Theory 27:657–677